en>Fustbariclation: // added link to idempotency

2015-01-03T05:14:09Z

// added link to idempotency

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	~~{{unreferenced\|date=March 2011}}~~		Nice to satisfy you, my title is Refugia. California is exactly where I've always been living and I adore every day residing here. Bookkeeping is what I do. To gather coins is a factor that I'm totally addicted to.<br><br>Feel free to visit my blog post; at home std test ([http://vei.cuaed.unam.mx/es/node/4882 mouse click the up coming post])
	~~In [[mathematics]]~~, ~~the~~ '~~''Fatou–Lebesgue theorem''' establishes a chain of [[inequality (mathematics)\|inequalities]] relating the [[integral]]s (in the sense of [[Lebesgue integration\|Lebesgue]]) of the [[limit superior~~ and limit inferior\|limit inferior]] and the [[limit superior and limit inferior\|limit superior]] of a [[sequence]] of [[function (mathematics)\|function]]s to the limit inferior and the limit superior of integrals of these functions. ~~The theorem~~ is ~~named after [[Pierre Fatou]] and [[Henri Léon Lebesgue]]~~.

	If the sequence of functions converges [[pointwise convergence\|pointwise]], the inequalities turn into [[equality (mathematics)\|equalities]] and the theorem reduces to Lebesgue's [[dominated convergence theorem]].

	~~==Statement of the theorem==~~
	Let ''f''<sub>1</sub>, ''f''<sub>2</sub>, ... denote a sequence of [[real number\|real]]-valued [[measurable function\|measurable]] functions defined on a [[measure space]] (''S'',''Σ'',''μ''). If there exists a ~~Lebesgue-integrable function ''g'' on ''S'' which dominates the sequence in absolute value, meaning~~ that \|''f''<sub>''n''</sub>\| ≤ ''g'' for all [[natural number]]s ''n'', then all ''f''<sub>''n''</sub> as well as the limit inferior and the limit superior of the ''f''<sub>''n''</sub> are integrable and
	~~:<math>~~
	~~\int_S \liminf_{n\to\infty} f_n\,d\mu~~
	~~\le \liminf_{n\to\infty} \int_S f_n\,d\mu~~
	~~\le \limsup_{n\to\infty} \int_S f_n\,d\mu~~
	~~\le \int_S \limsup_{n\to\infty} f_n\,d\mu\,.~~
	~~</math>~~
	~~Here the limit inferior and the limit superior of the ''f''<sub>''n''</sub> are taken pointwise. The integral of the absolute value of these limiting functions is bounded above by the integral of~~ '~~'g''.~~

	~~Since the middle inequality (for sequences of real numbers) is always true, the directions of the other inequalities are easy~~ to ~~remember~~.

	~~==Proof==~~
	~~All ''f''~~<~~sub~~>~~''n''~~<~~/sub~~> ~~as well as the limit inferior and the limit superior of the ''f''<sub>''n''</sub> are measurable and dominated in absolute value by ''g'', hence integrable.~~

	~~The first inequality follows by applying [[Fatou's lemma]]~~ to ~~the non-negative functions ''f''<sub>''n''</sub>&nbsp~~;+ ''g'' and using the [[Lebesgue_integral#Basic_theorems_of_the_Lebesgue_integral\|linearity of the Lebesgue integral]]. The last inequality is the [[Fatou's_lemma#Reverse_Fatou_lemma\|reverse Fatou lemma]].

	~~Since ''g'' also dominates the limit superior of the \|''f''<sub>''n''</sub>\|,~~

	~~:<math>0\le\biggl\|\int_S \liminf_{n\to\infty} f_n\,d\mu\biggr\|~~
	~~\le\int_S \Bigl\|\liminf_{n\to\infty} f_n\Bigr\|\,d\mu~~
	~~\le\int_S \limsup_{n\to\infty} \|f_n\|\,d\mu~~
	~~\le\int_S g\,d\mu</math>~~

	~~by the [[Lebesgue_integral#Basic_theorems_of_the_Lebesgue_integral\|monotonicity of the Lebesgue integral]]. The same estimates hold for the limit superior of the ''f''<sub>''n''</sub>.~~

	~~== References ==~~

	*[http://~~www.mat~~.~~univie~~.ac.at/~~~gerald~~/~~ftp~~/~~book-fa/index.html Topics in Real and Functional Analysis] by [[Gerald Teschl]], University of Vienna.~~

	~~==External links==~~
	*{{planetmath reference\|id=3679\|title=Fatou-Lebesgue theorem}}

	~~{{DEFAULTSORT:Fatou-Lebesgue theorem}}~~
	~~[[Category:Theorems in real analysis]]~~
	~~[[Category:Theorems in measure theory]]~~
	~~[[Category:Articles containing proofs]~~]

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en>Fustbariclation: // added link to idempotency

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